Inst. Phys. Conf. Ser. No 175
Paper presented at 7th Int. Conf. Computational Accelerator Physics, Michigan, USA, 15–18 October 2002
©2004 IOP Publishing Ltd
211
Muon beam ring cooler simulations using COSY INFINITY
Carlos O. Maidana†, Martin Berz† and Kyoko Makino‡
† Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824,
USA
‡ Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street,
Urbana, IL 61801-3080, USA
E-mail: [email protected], [email protected], [email protected]
Abstract. In this paper we present simulations using COSY INFINITY to study the behavior
of muon beams in a ring cooler designed by V. Balbekov[1]. Because of the substantial
transversal emittance, the nonlinearities play a very important limiting role that must be
understood and controlled well, which leads to the requirement of high order computation.
We describe the system, the approaches for the simulations of the large aperture solenoids and
magnetic sectors, and we show the nonlinear transfer maps as well as tracking simulations for
different field models, and compare with other methods based on various approximations.
PACS numbers: 02.60.Cb, 05.45.-a, 29.27.-a, 29.27.Eg, 41.85.-p, 41.85.Ja, 41.85.Lc
1. The ring and the simulation approaches
Various designs and ideas have been developed for cooling of short lived muon beams in
neutrino factories and muon colliders[2]. The concept of cooling is based on ionization
through material[3, 4, 5, 6], and to reduce cooling time, normally the system has a combined
structure, consisting of absorbing material, accelerating cavities and guiding magnets[2, 7].
Because of the huge transversal emittance of muon beams, the consideration of nonlinear
effects is an essential component in an earlier design stage. Lately, several designs of ring
coolers have been considered because of the ability to utilize cooling sections repeatedly, and
the additional potential for transversal and longitudinal emittance exchange. In this paper, we
analyze a muon beam ring cooler designed by V. Balbekov[1]. The layout of the system is
shown in Figure 1.
45
6.68 m
D .619 m
Bending magnet D 1.85 m
Solenoid coils
Direction of magnetic field
Liquid hydrogen absorber
LiH wedge absorber
205 MHz cavity
1.744 m
Figure 1. The layout of the tetra muon cooler designed by V. Balbekov[1].
212
198 cm
272 cm
198 cm
10.5 cm
3.5 cm
R 81 cm
J = 43.79 A/mm2
668 cm
Figure 2. The parameters of the long straight solenoidal section[1].
j = 65.73 164.34 -164.34 -65.73 A/mm^2
2.48 cm
4.95 cm
28.48
46.56 cm 38.16 38.16 46.56 cm
174.39 cm
Figure 3. The parameters of the short straight solenoidal section[1].
3
6
2
5
1
BZ(Tesla)
BZ(T)
4
3
2
0
−1
1
0
−400
−2
−200
0
Z (cm)
200
400
−3
−100
−50
0
Z (cm)
50
100
Figure 4. The hard edge model of the axial magnetic field in the long straight solenoidal
section (left) and in the short straight solenoidal section (right), assuming the solenoid coils
extend to infinity[1].
The ring consists of eight straight sections dominated by solenoids and eight
inhomogeneous bending magnets[1]. The four long straight sections have absorbing material
and accelerating cavities inside the solenoids, thus the aperture is very large. The parameters
of the solenoids in the long section are shown in Figure 2. The four short straight sections
have wedge absorbers to allow for transversal and longitudinal emittance exchange in the
middle, where the longitudinal magnetic field component flips direction. The parameters of
the solenoids in the short section are shown in Figure 3. Balbekov uses a hard edge model for
213
all magnets, and for simplicity of design purposes, it is usually adequate to assume that the
coil of the magnets extend to infinity[1]. The profiles of the longitudinal component of the
axial fields are shown in Figure 4.
First, if the length of the solenoids is finite, the field profiles differ significantly. Second,
in the long section, due to the huge aperture, the fringe field extension is exceedingly long[8].
Third, the hard edge model is somewhat unrealistic even for the case of ferromagnetic
yokes designed to block the fall-off of the fields[1] because they apparently need to have
an aperture large enough for passage of the muon beams that have large transversal emittance.
Considering these, we study the effects due to the different treatment of the fields using the
code COSY INFINITY[9]. The perhaps most realistic field treatment in the design stage is to
assume finitely long solenoids as indicated in Figures 2 and 3 without assuming the presence
of ferromagnetic yokes. Such field profiles are shown in Figures 5 and 6, including the outside
fringe regions. The code COSY INFINITY allows the nonlinear treatment of such solenoidal
fields including outside fringe field effects[9, 8]. For the purpose of comparison, we also
study the hard edge model of the fields. Balbekov uses the following linear kicks applied
to the transversal components of momentum to recover the most important edge field effect,
namely the induced overall rotation of the particles:[10]
C
C
(1)
∆px = Bz y, ∆py = − Bz x,
2
2
where p is in MeV/c, x and y are in meter, Bz is the longitudinal component of the axial field
at the edge in Tesla, and C = 299.79245. We also use the same linear kicks when the hard
edge model is used.
2. Transfer maps of solenoidal sections
We compare the effects of the different treatment of the solenoidal fields in the long and short
straight sections. A long section consists of three solenoidal parts, and a short section consists
of four solenoidal parts, with the longitudinal field flipping direction in the middle. Both the
long and the short sections are designed to have stronger current toward the middle. Due to
the flip of the field direction and the relatively small aperture, the short section is more readily
treatable by various approximations.
2.1. Short straight section
We compute the nonlinear transfer maps of the short section for different field models with
the beam kinetic energy of 250 MeV. We list the transfer maps of the hard edge model of
infinitely long solenoids first. For the purpose of comparison, we show the map without
and with the linear kicks (1). Below, parts of the nonlinear transfer maps are shown in the
notation of COSY INFINITY[9]. We observe that the linear x, a(= px /p0 ) terms and the
linear y, b(= py /p0 ) terms are almost decoupled when the kicks are applied, while they are
coupled without the presence of kicks. Thus, the linear kick approximation recovers the main
point of the linear motion and one of its important physical properties.
In the subsequent excerpts from transfer maps, the four columns represent final horizontal
position x (in meter), final horizontal slope a, final vertical position y (in meter), and final
vertical slope b, as a polynomial in the initial conditions. The exponents of the polynomial
are listed in the last column; for example, “4100” corresponds to the initial horizontal position
raised to the fourth power, and the initial horizontal slope raised to the first. The top lines of the
map represent the linear motion, and corresponds to the well-known transfer matrix (although
the latter is usually shown as the transpose of our format).
214
3
2
Bz (T)
1
0
-1
-2
-3
-2
-1
0
1
s (m)
2
3
Figure 5. The axial magnetic field profile of the short section with finitely long solenoids and
fringe fields.
Hard edge model with infinitely long solenoids (no linear kicks)
x_f
-0.1467136
1.015304
...
-15.71585
-36.02640
-38.39075
-33.14696
-14.27604
-7.567237
a_f
-1.796260
-0.1467136
...
27.23532
5.753202
16.29484
-7.116152
2.840011
-3.099152
y_f
b_f
0.9193886
-0.2657072
0.3060451E-11 0.9193886
...
...
-23.72338
-33.63860
-21.46530
-55.14127
-33.88081
-55.16902
-14.53667
-36.65477
-11.47237
-15.47623
-1.462962
-6.833978
xayb
1000
0100
....
5000
4100
3200
2300
1400
0500
Hard edge model with infinitely long solenoids with linear kicks
-0.1467136
1.015304
...
-2.968705
-12.74829
-22.23560
-22.28744
-15.60109
-7.567237
-0.9637261
-0.1467136
...
1.388108
-1.191336
-5.817135
-10.62269
-7.532831
-4.424200
-0.2022731E-03 0.5845774E-04
0.3060451E-11-0.2022730E-03
...
...
1.284507
1.099203
1.970436
4.189763
-0.2996768
5.761626
-3.195642
4.297247
-4.618497
1.461128
-1.462962
0.1989097E-01
1000
0100
....
5000
4100
3200
2300
1400
0500
We now list the same parts of the map of the hard edge model of finitely long solenoids.
As seen in Figure 5, the edge field strength is about half of that with infinitely long solenoids.
Thus, the map differs from the previous case, and the (x, x) and (a, a) terms show an obvious
difference.
Hard edge model with finitely long solenoids with linear kicks
0.3762572E-01-0.9144481
0.1324007E-03 0.9123892E-05
1.092008
0.3762572E-01 0.1216993E-10 0.1324008E-03
...
...
...
...
-4.559878
2.194351
1.925709
2.734348
-14.03917
1.790312
1.247951
8.100432
1000
0100
....
5000
4100
215
-26.15120
-29.55137
-20.98499
-8.419566
-0.5027561
-6.150214
-6.094498
-5.823382
-4.741542
-9.833677
-8.584871
-1.776100
9.421870
4.913186
-0.5031165
-0.7673755
3200
2300
1400
0500
We compare this map with the one computed for finitely long solenoids with correct
outside fringe field consideration without using the linear kicks. These two maps agree well,
confirming that the kick approach works well so far.
Finitely long solenoids with correct fringe field consideration
0.9113584E-02-0.9101484
0.2707143E-04-0.8741688E-06
1.098631
0.9113584E-02-0.1597370E-05 0.2707097E-04
...
...
...
...
-4.919054
1.095873
0.6296333
1.603693
-14.58589
0.6387705
-0.2037065
4.302134
-24.12332
-1.227331
-3.197581
3.442199
-26.81068
-7.088700
-4.268833
-0.1562313
-19.59752
-9.127604
-2.390249
-2.335453
-8.224955
-7.060173
0.1989972
-0.9178104
1000
0100
....
5000
4100
3200
2300
1400
0500
2.2. Long straight section
We performed the same study for the long straight section with the beam total energy of 250
MeV (the kinetic energy of 144.32 MeV). The hard edge model is used for infinitely long
solenoidal field, and for finitely long solenoidal field, where the edge field strength is again
about half of that with infinitely long solenoids. The computed transfer maps are compared
to the one with the correct outside fringe field consideration. Since no good agreement was
found between those three maps even in linear terms, we list only a part of the linear terms
below.
Hard edge model with infinitely long solenoids with linear kicks
0.7201144
-0.3623067
0.6140963
0.7201318
0.4256817
-0.2141606
0.3629942
0.4256523
1000
0100
Hard edge model with finitely long solenoids with linear kicks
-0.1764031E-02 0.4682935E-01-0.3214589E-01 0.9457639
-0.5217216E-01-0.1420209E-02 -1.053667
-0.3216291E-01
1000
0100
Finitely long solenoids with correct fringe field consideration
0.2334781
-1.157656
0.7546891
0.2462930
0.8859026E-01 0.2042114
-0.3132503
0.4123110E-01
1000
0100
The main reason for the disagreement between the hard edge model and the correct fringe
field treatment is due to the limitation of the linear kick approximation in (1). Comparing
the field profiles between the short section in Figure 5 and the long section in Figure 6,
the edge field strength is almost the same, namely about 1 Tesla, but the extension of the
outside fringe fields behaves differently. The outside fringe fields of the short section vanish
rapidly, but those of the long section cannot vanish even for a very long distance. By setting
the inner radius of the solenoids to 1/10 of the original radius while keeping all the other
parameters fixed, we computed the linear maps for the hard edge model and the correct fringe
field treatment for finitely long solenoids, and we found reasonable agreement.
216
6
5
Bz (T)
4
3
2
1
0
-2
0
2
4
6
8
s (m)
Figure 6. The axial magnetic field profile of the long section with finitely long solenoids and
fringe fields.
3
(sum of (linear map element)^2 ) /2
100*(Symplectic Error)
2.5
Difference
2
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Inner Radius / Original Inner radius (81cm)
0.9
1
Figure 7. The disagreement of the linear transfer maps between the hard edge model and the
correct fringe field treatment for the finitely long solenoids of the long section as functions of
the aperture.
Hard edge model with finitely long solenoids with linear kicks
(The inner radius is 1/10 of the original size.)
-0.1162300E-01-0.1163328E-01 0.2318096
0.9834308E-01-0.4517004E-02 -2.826397
0.3343426
0.2320568
1000
0100
Finitely long solenoids with correct fringe field consideration
(The inner radius is 1/10 of the original size.)
-0.5651780E-02-0.8030099E-02 0.2324166
0.6789750E-01-0.5511159E-02 -2.827323
0.3343818
0.2324200
1000
0100
Figure 7 shows the correlation of the agreement of the maps for various aperture sizes.
The difference in the sum of the square of linear terms and the difference in the symplectic
error are plotted as functions of the ratio of the inner radius to the original size.
217
3. Dynamics through the magnets in the ring
Since one half of the ring characterizes the whole system as seen in Figure 1, we compute
the transfer map of one half of the ring to study the beam dynamics through many revolutions
in the ring. Scanning the energies of the reference particle shows that under the presence
of dipole fringe fields and solenoid fringe fields, the linear motion is frequently unstable,
suggesting the need to re-fit the optical properties of the ring. To illustrate the performance,
we thus restricted ourselves to the design energy of 250 MeV total (=kinetic energy + muon
mass energy). Figures 8 and 9 show tracking for various cases. Figure 8 shows the hard edge
model of the solenoid with finitely long solenoids in the linear kick approximation. The left
picture shows tracking at order 9 with a hard edge model bending magnet, while the right
picture shows the effect of using a realistic bending magnet fringe field, which here leads
to unstable linear motion. Figure 9 shows the finitely long solenoids with correct fringe field
0.200
0.200
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9-th order, 100 1/2 turns, Etot=250 2k FR 0
9-th order, 100 1/2 turns, Etot=250 2k FR 2
Figure 8. Tracking 50 revolutions at reference energy of Etot =250 MeV with finitely long
solenoids in hard edge kick approximation, using hard edge dipole fields (left) and realistic
dipole fringe fields (right).
0.200
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7-th order, 100 1/2 turns, Etot=250 3s FR 0 EXPO
7-th order, 100 1/2 turns, Etot=250 3s FR 2 EXPO
Figure 9. Tracking 50 revolutions at reference energy of Etot =250 MeV with finitely long
solenoids with correct fringe field consideration, using hard edge dipole fields (left) and
realistic dipole fringe fields (right).
218
consideration; again the left picture shows the situation for a hard-edge bending magnet, while
the right picture shows the effects of a realistic bending magnet fringe field.
Particle tracking in the x-a phase space is done for 50 full revolutions in the ring, and the
Poincare sections are in the middle of the short straight solenoidal section, i.e. the upper left
corner of Figure 1. In Figures 8 and 9, the horizontal axis is the horizontal position x in meter,
and the vertical axis is the horizontal slope a = px /p0 . The particles with the initial positions
1, 2, ..., 7 cm are tracked in the ring, and the ends of lines showing the axes in the pictures are
±0.1 meter in x and ±0.2 in a.
Acknowledgments
We are thankful to V. Balbekov for providing all the necessary details for the simulation as
well as the pictures in Figures 1 through 4 to illustrate the system. The work was supported
by the Illinois Consortium for Accelerator Research, the US Department of Energy, and the
National Science Foundation.
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